Elastic properties closures using second-order homogenization theories: Case studies in composites of two isotropic constituents

Property closures delineate the complete set of feasible effective property combinations in a given composite material, and are of tremendous interest in highly constrained design applications. In recent papers, we have presented novel mathematical procedures that can successfully delineate theoretical elastic–plastic property closures based on first-order bounding theories for polycrystalline metals. This framework has been called microstructure sensitive design (MSD) and is rigorously grounded in spectral representation of invertible linkages between a statistical description of the microstructure (averaged over an ensemble) and its macroscale (effective) properties. In this paper, we have successfully extended this mathematical framework to include second-order homogenization theories that make predictions of the effective properties utilizing two-point spatial correlations in the microstructure. An idealized composite material system comprising two isotropic constituents, but exhibiting anisotropic effective properties at the macroscale (derived through placement of the constituents in the internal structure of the composite), was utilized in this study to demonstrate the viability of the proposed mathematical approaches. The development of the novel mathematical framework presented in this paper entailed the following essential steps: (1) critical validation of some of the variants of the second-order homogenization theory for the anisotropic effective elastic properties using finite element models followed by selection of a specific variant that is sufficiently accurate for moderate contrast problems while being amenable to be incorporated in the MSD framework, (2) transformation of the selected second-order homogenization theory into an efficient spectral representation that is ideally suited for exploration using quadratic or sequential quadratic programming, (3) use of Pareto optimal solution methods to delineate closures, and (4) development of novel strategies for identifying specific sets of microstructures corresponding to a selected combination of effective properties.